<!DOCTYPE html>
<html>
<head><meta name="generator" content="Hexo 3.9.0">
  <meta charset="utf-8">
  
  <title>hexo</title>
  <meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1">
  <meta property="og:type" content="website">
<meta property="og:title" content="hexo">
<meta property="og:url" content="http://yoursite.com/page/5/index.html">
<meta property="og:site_name" content="hexo">
<meta property="og:locale" content="default">
<meta name="twitter:card" content="summary">
<meta name="twitter:title" content="hexo">
  
    <link rel="alternate" href="/atom.xml" title="hexo" type="application/atom+xml">
  
  
    <link rel="icon" href="/favicon.ico">
  
  
    
  
  <link rel="stylesheet" href="/css/style.css">
  

</head>
</html>
<body>
  <div id="container">
    <div id="wrap">
      <header id="header">
  <div id="banner"></div>
  <div id="header-outer" class="outer">
    
    <div id="header-inner" class="inner">
      <nav id="sub-nav">
        
          <a id="nav-rss-link" class="nav-icon" href="/atom.xml" title="RSS Feed"></a>
        
        <a id="nav-search-btn" class="nav-icon" title="搜索"></a>
      </nav>
      <div id="search-form-wrap">
        <form action="//google.com/search" method="get" accept-charset="UTF-8" class="search-form"><input type="search" name="q" class="search-form-input" placeholder="Search"><button type="submit" class="search-form-submit">&#xF002;</button><input type="hidden" name="sitesearch" value="http://yoursite.com"></form>
      </div>
      <nav id="main-nav">
        <a id="main-nav-toggle" class="nav-icon"></a>
        
          <a class="main-nav-link" href="/">首页</a>
        
          <a class="main-nav-link" href="/archives">归档</a>
        
          <a class="main-nav-link" href="/categories/高等数学">高等数学</a>
        
          <a class="main-nav-link" href="/categories/线性代数">线性代数</a>
        
          <a class="main-nav-link" href="/categories/数据结构与算法">数据结构与算法</a>
        
          <a class="main-nav-link" href="/categories/English">英语</a>
        
          <a class="main-nav-link" href="/about">关于</a>
        
      </nav>
      
    </div>
    <div id="header-title" class="inner">
      <h1 id="logo-wrap">
        <a href="/" id="logo">hexo</a>
      </h1>
      
    </div>
  </div>
</header>
      <div class="outer">
        <section id="main">
  
    <article id="post-线性代数/实用大众线性代数/第4周矩阵运算的应用" class="article article-type-post" itemscope itemprop="blogPost">
  <div class="article-meta">
    <a href="/2019/05/17/线性代数/实用大众线性代数/第4周矩阵运算的应用/" class="article-date">
  <time datetime="2019-05-16T19:59:07.000Z" itemprop="datePublished">2019-05-17</time>
</a>
    
  <div class="article-category">
    <a class="article-category-link" href="/categories/线性代数/">线性代数</a>►<a class="article-category-link" href="/categories/线性代数/实用大众线性代数/">实用大众线性代数</a>
  </div>

  </div>
  <div class="article-inner">
    
    
      <header class="article-header">
        
  
    <h1 itemprop="name">
      <a class="article-title" href="/2019/05/17/线性代数/实用大众线性代数/第4周矩阵运算的应用/">第四周矩阵运算的应用</a>
    </h1>
  

      </header>
    
    <div class="article-entry" itemprop="articleBody">
      
        <!-- Table of Contents -->
        
        <h1 id="矩阵的转置"><a href="#矩阵的转置" class="headerlink" title="矩阵的转置"></a>矩阵的转置</h1><p>讲i行j列 转换为 j行i列 称为A的转置矩阵 $A^T$</p>
<script type="math/tex; mode=display">(A^T)^T = A</script><script type="math/tex; mode=display">(A+B)^T = A^T + B^T</script><script type="math/tex; mode=display">(\lambda A)^T = \lambda A^T</script><script type="math/tex; mode=display">(AB)^T = B^T A^T</script><p>若$A^T = A$,则A为对称矩阵</p>
<h1 id="矩阵的分块"><a href="#矩阵的分块" class="headerlink" title="矩阵的分块"></a>矩阵的分块</h1><p>矩阵拆分： A的子块<br>按行分块： 行向量<br>按列分块： 列向量</p>
<p>五大描述</p>
<h1 id="初等矩阵"><a href="#初等矩阵" class="headerlink" title="初等矩阵"></a>初等矩阵</h1><p>单位矩阵经过一次变换转换成的矩阵</p>
<script type="math/tex; mode=display">
A=\begin{bmatrix}
1&0&0\\
0&1&0\\
0&0&1\\
\end{bmatrix}
==R1,R2交换位置==>
\begin{bmatrix}
0&1&0\\
1&0&0\\
0&0&1\\
\end{bmatrix}</script><script type="math/tex; mode=display">
A=\begin{bmatrix}
1&0&0\\
0&1&0\\
0&0&1\\
\end{bmatrix}
==R3*5==>
\begin{bmatrix}
1&0&0\\
0&1&0\\
0&0&5\\
\end{bmatrix}</script><script type="math/tex; mode=display">
A=\begin{bmatrix}
1&0&0\\
0&1&0\\
0&0&1\\
\end{bmatrix}
=R1+5R2=>
\begin{bmatrix}
1&0&0\\
5&1&0\\
0&0&1\\
\end{bmatrix}</script><h2 id="以上三种对单位方阵的一次变换后的方阵统称为初等方阵Q"><a href="#以上三种对单位方阵的一次变换后的方阵统称为初等方阵Q" class="headerlink" title="以上三种对单位方阵的一次变换后的方阵统称为初等方阵Q"></a>以上三种对单位方阵的一次变换后的方阵统称为初等方阵Q</h2><h1 id="关于初等矩阵和方阵的定理"><a href="#关于初等矩阵和方阵的定理" class="headerlink" title="关于初等矩阵和方阵的定理"></a>关于初等矩阵和方阵的定理</h1><p>定理1：若A是一个矩阵，对A实行一次初等行变换，其结果等于在A的左边乘以相应的m阶初等矩阵Q</p>
<script type="math/tex; mode=display">
A\xrightarrow{初等行变换}B</script><script type="math/tex; mode=display">
QA = B</script><hr>
<p>定理2：<br>设A为N阶方阵，那么下面各命题等价，互为充要条件：</p>
<ol>
<li>A为可逆矩阵</li>
<li>线性方程组AX=0 只有零解</li>
<li>A可以通过有限次初等行变换变换为单位矩阵<br>$$<br>A=\begin{bmatrix}<br>1&amp;0&amp;0\<br>0&amp;1&amp;0\<br>0&amp;0&amp;1\<br>\end{bmatrix}</li>
</ol>
<p>B = \begin{bmatrix}<br>0\<br>0\<br>0\<br>\end{bmatrix}<br>只有零解</p>
<p>$$</p>
<ol>
<li>A可以表示为有限个初等矩阵的乘积</li>
</ol>

      
    </div>
    <footer class="article-footer">
      <a data-url="http://yoursite.com/2019/05/17/线性代数/实用大众线性代数/第4周矩阵运算的应用/" data-id="cjz7utmp4002fxhpegbb4gktv" class="article-share-link">分享</a>
      
      
      
  <ul class="article-tag-list"><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/矩阵乘法/">矩阵乘法</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/矩阵数乘/">矩阵数乘</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/矩阵线性变换/">矩阵线性变换</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/逆矩阵/">逆矩阵</a></li></ul>

    </footer>
  </div>
  
</article>
 


  
    <article id="post-线性代数/实用大众线性代数/第3周矩阵的四则运算" class="article article-type-post" itemscope itemprop="blogPost">
  <div class="article-meta">
    <a href="/2019/05/10/线性代数/实用大众线性代数/第3周矩阵的四则运算/" class="article-date">
  <time datetime="2019-05-10T02:16:09.000Z" itemprop="datePublished">2019-05-10</time>
</a>
    
  <div class="article-category">
    <a class="article-category-link" href="/categories/线性代数/">线性代数</a>►<a class="article-category-link" href="/categories/线性代数/实用大众线性代数/">实用大众线性代数</a>
  </div>

  </div>
  <div class="article-inner">
    
    
      <header class="article-header">
        
  
    <h1 itemprop="name">
      <a class="article-title" href="/2019/05/10/线性代数/实用大众线性代数/第3周矩阵的四则运算/">第三周矩阵的四则运算</a>
    </h1>
  

      </header>
    
    <div class="article-entry" itemprop="articleBody">
      
        <!-- Table of Contents -->
        
        <h1 id="矩阵运算及应用"><a href="#矩阵运算及应用" class="headerlink" title="矩阵运算及应用"></a>矩阵运算及应用</h1><h2 id="矩阵代数理论"><a href="#矩阵代数理论" class="headerlink" title="矩阵代数理论"></a>矩阵代数理论</h2><p>多个线性系统相互联结，构建更大，更复杂的系统，建立矩阵代数理论</p>
<h2 id="矩阵的加法"><a href="#矩阵的加法" class="headerlink" title="矩阵的加法"></a>矩阵的加法</h2><h3 id="超市上，下半年营业额"><a href="#超市上，下半年营业额" class="headerlink" title="超市上，下半年营业额"></a>超市上，下半年营业额</h3><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; A = randi([<span class="number">1</span> <span class="number">100</span>],<span class="number">5</span>,<span class="number">5</span>)</span><br><span class="line">A =</span><br><span class="line">      <span class="number">64</span>             <span class="number">54</span>             <span class="number">88</span>             <span class="number">42</span>             <span class="number">15</span></span><br><span class="line">      <span class="number">29</span>             <span class="number">45</span>             <span class="number">28</span>             <span class="number">21</span>             <span class="number">17</span></span><br><span class="line">      <span class="number">54</span>             <span class="number">13</span>             <span class="number">21</span>             <span class="number">95</span>             <span class="number">63</span></span><br><span class="line">      <span class="number">70</span>             <span class="number">50</span>             <span class="number">57</span>              <span class="number">9</span>             <span class="number">58</span></span><br><span class="line">      <span class="number">50</span>             <span class="number">86</span>             <span class="number">65</span>             <span class="number">11</span>              <span class="number">6</span></span><br><span class="line">&gt;&gt; B = randi([<span class="number">1</span> <span class="number">100</span>],<span class="number">5</span>,<span class="number">5</span>)</span><br><span class="line">B =</span><br><span class="line">      <span class="number">94</span>             <span class="number">94</span>             <span class="number">18</span>             <span class="number">31</span>              <span class="number">3</span></span><br><span class="line">      <span class="number">73</span>             <span class="number">99</span>             <span class="number">40</span>             <span class="number">30</span>             <span class="number">85</span></span><br><span class="line">      <span class="number">74</span>             <span class="number">86</span>             <span class="number">14</span>             <span class="number">34</span>             <span class="number">56</span></span><br><span class="line">       <span class="number">7</span>             <span class="number">79</span>              <span class="number">4</span>             <span class="number">47</span>             <span class="number">86</span></span><br><span class="line">      <span class="number">87</span>             <span class="number">52</span>             <span class="number">94</span>             <span class="number">65</span>             <span class="number">35</span></span><br><span class="line">&gt;&gt; A+B</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">     <span class="number">158</span>            <span class="number">148</span>            <span class="number">106</span>             <span class="number">73</span>             <span class="number">18</span></span><br><span class="line">     <span class="number">102</span>            <span class="number">144</span>             <span class="number">68</span>             <span class="number">51</span>            <span class="number">102</span></span><br><span class="line">     <span class="number">128</span>             <span class="number">99</span>             <span class="number">35</span>            <span class="number">129</span>            <span class="number">119</span></span><br><span class="line">      <span class="number">77</span>            <span class="number">129</span>             <span class="number">61</span>             <span class="number">56</span>            <span class="number">144</span></span><br><span class="line">     <span class="number">137</span>            <span class="number">138</span>            <span class="number">159</span>             <span class="number">76</span>             <span class="number">41</span></span><br></pre></td></tr></table></figure>
<h2 id="矩阵的数乘"><a href="#矩阵的数乘" class="headerlink" title="矩阵的数乘"></a>矩阵的数乘</h2><h3 id="平时成绩和期末成绩占比"><a href="#平时成绩和期末成绩占比" class="headerlink" title="平时成绩和期末成绩占比"></a>平时成绩和期末成绩占比</h3><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; <span class="number">0.3</span>*A+<span class="number">0.7</span>*B</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">      <span class="number">85</span>             <span class="number">82</span>             <span class="number">39</span>            <span class="number">343</span>/<span class="number">10</span>          <span class="number">33</span>/<span class="number">5</span></span><br><span class="line">     <span class="number">299</span>/<span class="number">5</span>          <span class="number">414</span>/<span class="number">5</span>          <span class="number">182</span>/<span class="number">5</span>          <span class="number">273</span>/<span class="number">10</span>         <span class="number">323</span>/<span class="number">5</span></span><br><span class="line">      <span class="number">68</span>            <span class="number">641</span>/<span class="number">10</span>         <span class="number">161</span>/<span class="number">10</span>         <span class="number">523</span>/<span class="number">10</span>         <span class="number">581</span>/<span class="number">10</span></span><br><span class="line">     <span class="number">259</span>/<span class="number">10</span>         <span class="number">703</span>/<span class="number">10</span>         <span class="number">199</span>/<span class="number">10</span>         <span class="number">178</span>/<span class="number">5</span>          <span class="number">388</span>/<span class="number">5</span></span><br><span class="line">     <span class="number">759</span>/<span class="number">10</span>         <span class="number">311</span>/<span class="number">5</span>          <span class="number">853</span>/<span class="number">10</span>         <span class="number">244</span>/<span class="number">5</span>          <span class="number">263</span>/<span class="number">10</span></span><br><span class="line">&gt;&gt; A = <span class="built_in">ones</span>(<span class="number">5</span>)</span><br><span class="line">A =</span><br><span class="line">       <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span></span><br><span class="line">       <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span></span><br><span class="line">       <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span></span><br><span class="line">       <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span></span><br><span class="line">       <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span></span><br><span class="line">&gt;&gt; <span class="number">5</span>*A</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">       <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span></span><br><span class="line">       <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span></span><br><span class="line">       <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span></span><br><span class="line">       <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span></span><br><span class="line">       <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span></span><br><span class="line"> &gt;&gt; A+A</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">       <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span></span><br><span class="line">       <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span></span><br><span class="line">       <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span></span><br><span class="line">       <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span></span><br><span class="line">       <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span></span><br><span class="line"></span><br><span class="line">&gt;&gt; A+A+<span class="number">4</span>*A</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">       <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span></span><br><span class="line">       <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span></span><br><span class="line">       <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span></span><br><span class="line">       <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span></span><br><span class="line">       <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span></span><br><span class="line">&gt;&gt;</span><br></pre></td></tr></table></figure>
<h2 id="矩阵的线性运算"><a href="#矩阵的线性运算" class="headerlink" title="矩阵的线性运算"></a>矩阵的线性运算</h2><div class="table-container">
<table>
<thead>
<tr>
<th>规则</th>
<th>表达式</th>
</tr>
</thead>
<tbody>
<tr>
<td>交换律</td>
<td>A+B = B+A</td>
</tr>
<tr>
<td>结合律</td>
<td>A+(B+C) = (A+B) +C</td>
</tr>
<tr>
<td>数乘结合律</td>
<td>($\lambda <em> \mu$) A = $\lambda$ </em> ( $\mu$ A)</td>
</tr>
<tr>
<td>数乘分配律</td>
<td>$\lambda$ (A+B)  = $\lambda$ A + $\lambda$ B</td>
</tr>
</tbody>
</table>
</div>
<h1 id="矩阵的乘法"><a href="#矩阵的乘法" class="headerlink" title="矩阵的乘法"></a>矩阵的乘法</h1><h2 id="多服装厂协同生产"><a href="#多服装厂协同生产" class="headerlink" title="多服装厂协同生产"></a>多服装厂协同生产</h2><div class="table-container">
<table>
<thead>
<tr>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td>单位产量/厂名</td>
<td>甲</td>
<td>乙</td>
<td>丙</td>
<td>丁</td>
</tr>
<tr>
<td>帽子</td>
<td>20</td>
<td>4</td>
<td>2</td>
<td>7</td>
</tr>
<tr>
<td>上衣</td>
<td>10</td>
<td>18</td>
<td>5</td>
<td>6</td>
</tr>
<tr>
<td>裤子</td>
<td>5</td>
<td>7</td>
<td>16</td>
<td>3</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>生产时长</td>
<td>8</td>
<td>10</td>
<td>5</td>
<td>9</td>
</tr>
</tbody>
</table>
</div>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; A</span><br><span class="line">A =</span><br><span class="line">      <span class="number">20</span>              <span class="number">4</span>              <span class="number">2</span>              <span class="number">7</span></span><br><span class="line">      <span class="number">10</span>             <span class="number">18</span>              <span class="number">5</span>              <span class="number">6</span></span><br><span class="line">       <span class="number">5</span>              <span class="number">7</span>             <span class="number">16</span>              <span class="number">3</span></span><br><span class="line">&gt;&gt; b</span><br><span class="line">b =</span><br><span class="line">       <span class="number">8</span></span><br><span class="line">      <span class="number">10</span></span><br><span class="line">       <span class="number">5</span></span><br><span class="line">       <span class="number">9</span></span><br><span class="line">&gt;&gt; A*b</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">     <span class="number">273</span></span><br><span class="line">     <span class="number">339</span></span><br><span class="line">     <span class="number">217</span></span><br></pre></td></tr></table></figure>
<h2 id="矩阵乘积-定义"><a href="#矩阵乘积-定义" class="headerlink" title="矩阵乘积 定义"></a>矩阵乘积 定义</h2><p>设A 是mxs 矩阵 ，B是 sxn 矩阵，则 AxB 为 mxn的矩阵C， C的各个元素为：</p>
<script type="math/tex; mode=display">
C_{ij} =  \sum_{k=1}^s a_{ik}b_{kj} = a_{i1}*b_{1j} +a_{i2}*b_{j2}+ \cdots+a_{is}*b_{sj}
(i = 1,2,\cdots, m) (j = 1,2, \cdots, n)</script><p>记作：</p>
<script type="math/tex; mode=display">
C = A B</script><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">可乘条件： A的列数 = B的行数</span><br><span class="line">AB乘积的形状： A行 ， B列</span><br><span class="line">AB乘积元素的构成： A的行与B的列的内积</span><br></pre></td></tr></table></figure>
<h2 id="线性变换"><a href="#线性变换" class="headerlink" title="线性变换"></a>线性变换</h2><script type="math/tex; mode=display">
变量 Y = [y_1, y_2 , \cdots, y_m],能够由X = [x_1, x_2, \cdots, x_n] 线性表示，
则成为此变量X到变量Y的线性变换</script><p>多次线性变换等价于矩阵相乘</p>
<p>若 Y = AX ， X = BT，则 Y= ABT</p>
<p>方程组的矩阵表示</p>
<p>A = [ 1, 2, 3]    B = [4; 5; 6]      AB  $ \neq $ BA</p>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; A = [ <span class="number">1</span>, <span class="number">2</span>, <span class="number">3</span>]</span><br><span class="line">A =</span><br><span class="line">       <span class="number">1</span>              <span class="number">2</span>              <span class="number">3</span></span><br><span class="line">&gt;&gt; B = [<span class="number">4</span>; <span class="number">5</span>; <span class="number">6</span>]</span><br><span class="line">B =</span><br><span class="line">       <span class="number">4</span></span><br><span class="line">       <span class="number">5</span></span><br><span class="line">       <span class="number">6</span></span><br><span class="line">&gt;&gt; A*B</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">      <span class="number">32</span></span><br><span class="line">&gt;&gt;</span><br><span class="line">&gt;&gt;</span><br><span class="line">&gt;&gt; B*A</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">       <span class="number">4</span>              <span class="number">8</span>             <span class="number">12</span></span><br><span class="line">       <span class="number">5</span>             <span class="number">10</span>             <span class="number">15</span></span><br><span class="line">       <span class="number">6</span>             <span class="number">12</span>             <span class="number">18</span></span><br></pre></td></tr></table></figure>
<p>空间次序不能交换</p>
<ol>
<li>AB C = A （BC）</li>
<li>A （B+C） = AB + AC<br>3.（A + B）C = AC + BC</li>
</ol>
<script type="math/tex; mode=display">A_{mn}I_{n} = I_mA_{mn}</script><p>5.A，B均为上（下）三角矩阵，则C = AB 也为上（下）三角矩阵</p>
<h1 id="矩阵的逆（互为倒数）"><a href="#矩阵的逆（互为倒数）" class="headerlink" title="矩阵的逆（互为倒数）"></a>矩阵的逆（互为倒数）</h1><p>引例：</p>
<p>xy = 1， 则x，y互为倒数</p>
<p>若 X Y = I 则 X Y 互逆</p>
<p>Y = AX 转换成  X = A’ Y</p>
<ol>
<li>A ， $\ A^{-1}$ 逆矩阵唯一</li>
<li>若A B 同阶可逆方阵，满足AB = I ，则BA = I  （AB互逆）</li>
<li>若A可逆，则 $\ A^{-1}$ 也可逆，$\ ( A^{-1})^{-1}$ = A</li>
<li>若A可逆, $\lambda \neq 0$ ,则 $\lambda$ A 可逆</li>
<li>$(AB)^{-1} = B^{-1} A^{-1}$</li>
</ol>
<h2 id="逆矩阵看做-矩阵除法"><a href="#逆矩阵看做-矩阵除法" class="headerlink" title="逆矩阵看做 矩阵除法"></a>逆矩阵看做 矩阵除法</h2><p>若</p>
<script type="math/tex; mode=display">AB = C</script><p>则</p>
<script type="math/tex; mode=display">B = A^{-1}C</script><script type="math/tex; mode=display">A = CB^{-1}</script>
      
    </div>
    <footer class="article-footer">
      <a data-url="http://yoursite.com/2019/05/10/线性代数/实用大众线性代数/第3周矩阵的四则运算/" data-id="cjz7utmp3002bxhpelsvwkkc6" class="article-share-link">分享</a>
      
      
      
  <ul class="article-tag-list"><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/矩阵乘法/">矩阵乘法</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/矩阵数乘/">矩阵数乘</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/矩阵线性变换/">矩阵线性变换</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/逆矩阵/">逆矩阵</a></li></ul>

    </footer>
  </div>
  
</article>
 


  
    <article id="post-线性代数/实用大众线性代数/第2周用MATLAB解线性方程组" class="article article-type-post" itemscope itemprop="blogPost">
  <div class="article-meta">
    <a href="/2019/01/16/线性代数/实用大众线性代数/第2周用MATLAB解线性方程组/" class="article-date">
  <time datetime="2019-01-16T08:20:44.000Z" itemprop="datePublished">2019-01-16</time>
</a>
    
  <div class="article-category">
    <a class="article-category-link" href="/categories/线性代数/">线性代数</a>►<a class="article-category-link" href="/categories/线性代数/实用大众线性代数/">实用大众线性代数</a>
  </div>

  </div>
  <div class="article-inner">
    
    
      <header class="article-header">
        
  
    <h1 itemprop="name">
      <a class="article-title" href="/2019/01/16/线性代数/实用大众线性代数/第2周用MATLAB解线性方程组/">第二周用MATLAB解线性方程组</a>
    </h1>
  

      </header>
    
    <div class="article-entry" itemprop="articleBody">
      
        <!-- Table of Contents -->
        
        <h1 id="MATLAB语言概述"><a href="#MATLAB语言概述" class="headerlink" title="MATLAB语言概述"></a>MATLAB语言概述</h1><h2 id="科学计算工具"><a href="#科学计算工具" class="headerlink" title="科学计算工具"></a>科学计算工具</h2><div class="table-container">
<table>
<thead>
<tr>
<th>年代</th>
<th>计算工具</th>
</tr>
</thead>
<tbody>
<tr>
<td>东汉（220年左右）</td>
<td>算盘</td>
</tr>
<tr>
<td>1630年</td>
<td>计算尺</td>
</tr>
<tr>
<td>1970年</td>
<td>计算器</td>
</tr>
<tr>
<td>1980年开始</td>
<td>计算机及科学计算语言 MATLAB</td>
</tr>
</tbody>
</table>
</div>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">线性代数 是计算机显示其优越性的最早的发源地</span><br><span class="line">不要墨守经典，无视现代。</span><br><span class="line">成为时代进步的实践者和宣传员</span><br></pre></td></tr></table></figure>
<h2 id="MATLAB语言特点"><a href="#MATLAB语言特点" class="headerlink" title="MATLAB语言特点"></a>MATLAB语言特点</h2><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">MATLAB：MATrix LABoratory </span><br><span class="line">以矩阵运算为基础的交互式程序语言</span><br></pre></td></tr></table></figure>
<ol>
<li><p>起点高</p>
</li>
<li><p>UI</p>
</li>
<li><p>作图功能</p>
</li>
<li><p>功能和可扩展性</p>
</li>
</ol>
<hr>
<h1 id="基本语法"><a href="#基本语法" class="headerlink" title="基本语法"></a>基本语法</h1><ol>
<li>标识符</li>
<li>矩阵及其元素赋值</li>
<li>复数</li>
<li>变量检车</li>
<li>基本赋值矩阵</li>
</ol>
<h2 id="标识符"><a href="#标识符" class="headerlink" title="标识符"></a>标识符</h2><p>大小写区别， 首字符英文<br>双精度的64位 （8个字节）10的+- 308次幂</p>
<h2 id="矩阵赋值"><a href="#矩阵赋值" class="headerlink" title="矩阵赋值"></a>矩阵赋值</h2><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line">A = [<span class="number">1</span> <span class="number">2</span> <span class="number">3</span>; <span class="number">4</span> <span class="number">5</span> <span class="number">6</span>; <span class="number">7</span> <span class="number">8</span> <span class="number">9</span>]</span><br><span class="line">A =</span><br><span class="line"></span><br><span class="line">     <span class="number">1</span>     <span class="number">2</span>     <span class="number">3</span></span><br><span class="line">     <span class="number">4</span>     <span class="number">5</span>     <span class="number">6</span></span><br><span class="line">     <span class="number">7</span>     <span class="number">8</span>     <span class="number">9</span></span><br></pre></td></tr></table></figure>
<h2 id="元素赋值"><a href="#元素赋值" class="headerlink" title="元素赋值"></a>元素赋值</h2><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br></pre></td><td class="code"><pre><span class="line">A(<span class="number">4</span>,<span class="number">3</span>) = <span class="number">6</span></span><br><span class="line">A =</span><br><span class="line"></span><br><span class="line">     <span class="number">1</span>     <span class="number">2</span>     <span class="number">3</span></span><br><span class="line">     <span class="number">4</span>     <span class="number">5</span>     <span class="number">6</span></span><br><span class="line">     <span class="number">7</span>     <span class="number">8</span>     <span class="number">9</span></span><br><span class="line">     <span class="number">0</span>     <span class="number">0</span>     <span class="number">6</span></span><br><span class="line">     <span class="number">6</span>     <span class="number">6</span>     <span class="number">6</span></span><br><span class="line">     </span><br><span class="line"></span><br><span class="line">A(<span class="number">5</span>,:) = [ <span class="number">6</span> <span class="number">6</span> <span class="number">6</span> ]</span><br><span class="line">A =</span><br><span class="line"></span><br><span class="line">     <span class="number">1</span>     <span class="number">2</span>     <span class="number">3</span></span><br><span class="line">     <span class="number">4</span>     <span class="number">5</span>     <span class="number">6</span></span><br><span class="line">     <span class="number">7</span>     <span class="number">8</span>     <span class="number">9</span></span><br><span class="line">     <span class="number">0</span>     <span class="number">0</span>     <span class="number">0</span></span><br><span class="line">     <span class="number">6</span>     <span class="number">6</span>     <span class="number">6</span></span><br></pre></td></tr></table></figure>
<h2 id="元素提取"><a href="#元素提取" class="headerlink" title="元素提取"></a>元素提取</h2><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line">b = A([<span class="number">2</span>, <span class="number">4</span>], [<span class="number">1</span>, <span class="number">3</span>]) <span class="comment">% (行, 列) %</span></span><br><span class="line">b =</span><br><span class="line"></span><br><span class="line">     <span class="number">4</span>     <span class="number">6</span></span><br><span class="line">     <span class="number">0</span>     <span class="number">6</span></span><br></pre></td></tr></table></figure>
<h2 id="元素的删除"><a href="#元素的删除" class="headerlink" title="元素的删除"></a>元素的删除</h2><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line">A([<span class="number">2</span>, <span class="number">4</span>, <span class="number">5</span>], :) = [] <span class="comment">% 2 4 5 行 删除</span></span><br><span class="line">A =</span><br><span class="line"></span><br><span class="line">     <span class="number">1</span>     <span class="number">2</span>     <span class="number">3</span></span><br><span class="line">     <span class="number">7</span>     <span class="number">8</span>     <span class="number">9</span></span><br></pre></td></tr></table></figure>
<h2 id="常数乘法"><a href="#常数乘法" class="headerlink" title="常数乘法"></a>常数乘法</h2><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line">A*<span class="number">2</span>               <span class="comment">%系数乘法%</span></span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line"></span><br><span class="line">     <span class="number">2</span>     <span class="number">4</span>     <span class="number">6</span></span><br><span class="line">    <span class="number">14</span>    <span class="number">16</span>    <span class="number">18</span></span><br></pre></td></tr></table></figure>
<h2 id="矩阵转置"><a href="#矩阵转置" class="headerlink" title="矩阵转置"></a>矩阵转置</h2><p>M行N列  下标交换位置<br><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line">A'              <span class="comment">%转置%</span></span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line"></span><br><span class="line">     <span class="number">1</span>     <span class="number">7</span></span><br><span class="line">     <span class="number">2</span>     <span class="number">8</span></span><br><span class="line">     <span class="number">3</span>     <span class="number">9</span></span><br></pre></td></tr></table></figure></p>
<hr>
<h1 id="特殊矩阵"><a href="#特殊矩阵" class="headerlink" title="特殊矩阵"></a>特殊矩阵</h1><h2 id="赋值函数"><a href="#赋值函数" class="headerlink" title="赋值函数"></a>赋值函数</h2><div class="table-container">
<table>
<thead>
<tr>
<th>函数名称</th>
<th>描述</th>
</tr>
</thead>
<tbody>
<tr>
<td>zeros(n, m)</td>
<td>全0</td>
</tr>
<tr>
<td>ones(n, m)</td>
<td>全1</td>
</tr>
<tr>
<td>rand(n,m)</td>
<td>随机数矩阵</td>
</tr>
<tr>
<td>randn(n,m)</td>
<td>正态随机矩阵</td>
</tr>
<tr>
<td>eye(n)</td>
<td>单位矩阵</td>
</tr>
</tbody>
</table>
</div>
<h2 id="变量检查-who-whos"><a href="#变量检查-who-whos" class="headerlink" title="变量检查 who whos"></a>变量检查 who whos</h2><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br></pre></td><td class="code"><pre><span class="line">pause,f5=<span class="built_in">linspace</span>(<span class="number">0</span>,<span class="number">1</span>,<span class="number">5</span>)</span><br><span class="line">f5 =</span><br><span class="line">         <span class="number">0</span>    <span class="number">0.2500</span>    <span class="number">0.5000</span>    <span class="number">0.7500</span>    <span class="number">1.0000</span></span><br><span class="line">pause,fb1=[f1,f3;f4,f2]</span><br><span class="line">fb1 =</span><br><span class="line">     <span class="number">1</span>     <span class="number">1</span>     <span class="number">8</span>     <span class="number">1</span>     <span class="number">6</span></span><br><span class="line">     <span class="number">1</span>     <span class="number">1</span>     <span class="number">3</span>     <span class="number">5</span>     <span class="number">7</span></span><br><span class="line">     <span class="number">1</span>     <span class="number">1</span>     <span class="number">4</span>     <span class="number">9</span>     <span class="number">2</span></span><br><span class="line">     <span class="number">1</span>     <span class="number">0</span>     <span class="number">0</span>     <span class="number">0</span>     <span class="number">0</span></span><br><span class="line">     <span class="number">0</span>     <span class="number">1</span>     <span class="number">0</span>     <span class="number">0</span>     <span class="number">0</span></span><br><span class="line">pause,fb2=[fb1;f5]</span><br><span class="line">fb2 =</span><br><span class="line">    <span class="number">1.0000</span>    <span class="number">1.0000</span>    <span class="number">8.0000</span>    <span class="number">1.0000</span>    <span class="number">6.0000</span></span><br><span class="line">    <span class="number">1.0000</span>    <span class="number">1.0000</span>    <span class="number">3.0000</span>    <span class="number">5.0000</span>    <span class="number">7.0000</span></span><br><span class="line">    <span class="number">1.0000</span>    <span class="number">1.0000</span>    <span class="number">4.0000</span>    <span class="number">9.0000</span>    <span class="number">2.0000</span></span><br><span class="line">    <span class="number">1.0000</span>         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span></span><br><span class="line">         <span class="number">0</span>    <span class="number">1.0000</span>         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span></span><br><span class="line">         <span class="number">0</span>    <span class="number">0.2500</span>    <span class="number">0.5000</span>    <span class="number">0.7500</span>    <span class="number">1.0000</span></span><br><span class="line">pause,f=[<span class="number">0.000073</span> <span class="number">5.33e-6</span>]</span><br><span class="line">f =</span><br><span class="line">   <span class="number">1.0e-04</span> *</span><br><span class="line">    <span class="number">0.7300</span>    <span class="number">0.0533</span></span><br><span class="line">pause,<span class="built_in">disp</span>(<span class="string">' 变量的检查'</span>)</span><br><span class="line"> 变量的检查</span><br><span class="line">pause,who</span><br><span class="line"></span><br><span class="line">Your variables are:</span><br><span class="line"></span><br><span class="line">A        U1       b        f1       f4       fb2      l1       logoax   x1       z        </span><br><span class="line">L        a        c        f2       f5       g        l2       s        x2       </span><br><span class="line">U0       <span class="built_in">ans</span>      f        f3       fb1      ip       logoFig  x        y        </span><br><span class="line"></span><br><span class="line"></span><br><span class="line">pause,whos</span><br><span class="line">  Name          Size            Bytes  Class                                Attributes</span><br><span class="line"></span><br><span class="line">  A             <span class="number">3</span>x4                <span class="number">96</span>  double                                         </span><br><span class="line">  L            <span class="number">51</span>x51            <span class="number">20808</span>  double                                         </span><br><span class="line">  U0            <span class="number">3</span>x4                <span class="number">96</span>  double                                         </span><br><span class="line">  U1            <span class="number">3</span>x4                <span class="number">96</span>  double                                         </span><br><span class="line">  a             <span class="number">4</span>x3                <span class="number">96</span>  double                                         </span><br><span class="line">  <span class="built_in">ans</span>           <span class="number">5</span>x5               <span class="number">200</span>  double                                         </span><br><span class="line">  b             <span class="number">2</span>x2                <span class="number">32</span>  double                                         </span><br><span class="line">  c             <span class="number">1</span>x1                <span class="number">16</span>  double                               <span class="built_in">complex</span>   </span><br><span class="line">  f             <span class="number">1</span>x2                <span class="number">16</span>  double                                         </span><br><span class="line">  f1            <span class="number">3</span>x2                <span class="number">48</span>  double                                         </span><br><span class="line">  f2            <span class="number">2</span>x3                <span class="number">48</span>  double                                         </span><br><span class="line">  f3            <span class="number">3</span>x3                <span class="number">72</span>  double                                         </span><br><span class="line">  f4            <span class="number">2</span>x2                <span class="number">32</span>  double                                         </span><br><span class="line">  f5            <span class="number">1</span>x5                <span class="number">40</span>  double                                         </span><br><span class="line">  fb1           <span class="number">5</span>x5               <span class="number">200</span>  double                                         </span><br><span class="line">  fb2           <span class="number">6</span>x5               <span class="number">240</span>  double                                         </span><br><span class="line">  g             <span class="number">1</span>x1                <span class="number">16</span>  double                               <span class="built_in">complex</span>   </span><br><span class="line">  ip            <span class="number">1</span>x3                <span class="number">24</span>  double                                         </span><br><span class="line">  l1            <span class="number">1</span>x1                 <span class="number">0</span>  matlab.graphics.primitive.Light                </span><br><span class="line">  l2            <span class="number">1</span>x1                 <span class="number">0</span>  matlab.graphics.primitive.Light                </span><br><span class="line">  logoFig       <span class="number">1</span>x1                 <span class="number">0</span>  matlab.ui.Figure                               </span><br><span class="line">  logoax        <span class="number">1</span>x1                 <span class="number">0</span>  matlab.graphics.axis.Axes                      </span><br><span class="line">  s             <span class="number">1</span>x1                 <span class="number">0</span>  matlab.graphics.primitive.Surface              </span><br><span class="line">  x             <span class="number">1</span>x5                <span class="number">40</span>  double                                         </span><br><span class="line">  x1            <span class="number">1</span>x1                 <span class="number">8</span>  double                                         </span><br><span class="line">  x2            <span class="number">1</span>x1                 <span class="number">8</span>  double                                         </span><br><span class="line">  y             <span class="number">1</span>x1                 <span class="number">8</span>  double                                         </span><br><span class="line">  z             <span class="number">2</span>x2                <span class="number">64</span>  double                               <span class="built_in">complex</span></span><br></pre></td></tr></table></figure>

      
    </div>
    <footer class="article-footer">
      <a data-url="http://yoursite.com/2019/01/16/线性代数/实用大众线性代数/第2周用MATLAB解线性方程组/" data-id="cjz7utmp30029xhpebgpcu4vx" class="article-share-link">分享</a>
      
      
      
  <ul class="article-tag-list"><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/MATLAB/">MATLAB</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/单位矩阵/">单位矩阵</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/矩阵转置/">矩阵转置</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/计算工具/">计算工具</a></li></ul>

    </footer>
  </div>
  
</article>
 


  
    <article id="post-线性代数/实用大众线性代数/第1周线性方程组与矩阵" class="article article-type-post" itemscope itemprop="blogPost">
  <div class="article-meta">
    <a href="/2019/01/14/线性代数/实用大众线性代数/第1周线性方程组与矩阵/" class="article-date">
  <time datetime="2019-01-14T10:02:08.000Z" itemprop="datePublished">2019-01-14</time>
</a>
    
  <div class="article-category">
    <a class="article-category-link" href="/categories/线性代数/">线性代数</a>►<a class="article-category-link" href="/categories/线性代数/实用大众线性代数/">实用大众线性代数</a>
  </div>

  </div>
  <div class="article-inner">
    
    
      <header class="article-header">
        
  
    <h1 itemprop="name">
      <a class="article-title" href="/2019/01/14/线性代数/实用大众线性代数/第1周线性方程组与矩阵/">第一周线性方程组与矩阵</a>
    </h1>
  

      </header>
    
    <div class="article-entry" itemprop="articleBody">
      
        <!-- Table of Contents -->
        
        <h1 id="高斯消元法"><a href="#高斯消元法" class="headerlink" title="高斯消元法"></a>高斯消元法</h1><h2 id="N元线性方程组"><a href="#N元线性方程组" class="headerlink" title="N元线性方程组"></a>N元线性方程组</h2><p>设m为方程数,n为变元数, 则n元方程组可表示为: （n &gt; m）</p>
<script type="math/tex; mode=display">
m组\begin{cases}
a_{11}x_1 + a_{12}x_{2} + \cdots\cdots + a_{1n}x_{n} &= &b_1 \\
a_{21}x_1  +a_{22}x_{2} +\cdots \cdots + a_{2n}x_{n} & = &b_2  \\
\vdots & \\
\vdots &  \\
\underbrace { a_{m1}x_1 + a_{m2}x_{2} +  \cdots\cdots  +  a_{mn}x_{n} = b_m }_{n个}\\
\end{cases} \tag{n元线性方程组}</script><h2 id="高斯消元法-1"><a href="#高斯消元法-1" class="headerlink" title="高斯消元法"></a>高斯消元法</h2><h3 id="消元过程"><a href="#消元过程" class="headerlink" title="消元过程"></a>消元过程</h3><p>通过 同解变换  转换为 阶梯形 同解方程组</p>
<h3 id="三种同解变换"><a href="#三种同解变换" class="headerlink" title="三种同解变换"></a>三种同解变换</h3><p>位置变换:  交换方程组位置</p>
<p>数乘变换:  方程左右两边 同乘 常数K</p>
<p>消元变换: 方程某元的K倍加到另一个方程上 消掉该元</p>
<h1 id="矩阵及矩阵的初等变换"><a href="#矩阵及矩阵的初等变换" class="headerlink" title="矩阵及矩阵的初等变换"></a>矩阵及矩阵的初等变换</h1><h2 id="矩阵的由来"><a href="#矩阵的由来" class="headerlink" title="矩阵的由来"></a>矩阵的由来</h2><p>提取方程组的系数 获得 系数数表  常数数表</p>
<script type="math/tex; mode=display">
A=\begin{bmatrix}
1&2&3\\
4&5&6\\
7&8&9\\
\end{bmatrix}
b=\begin{bmatrix}
1\\
4\\
7\\
\end{bmatrix}
\tag {方程组的系数矩阵和常数矩阵}</script><h2 id="矩阵的定义"><a href="#矩阵的定义" class="headerlink" title="矩阵的定义"></a>矩阵的定义</h2><p>由m x n个数 构成的m行n列的矩形数表</p>
<script type="math/tex; mode=display">
A=\begin{bmatrix}
a_{11}& a_{12} & \dots& a_{1n}\\
a_{21}& a_{22} & \dots& a_{2n}\\
\vdots& \vdots & \ddots&\vdots\\
a_{m1}& a_{m2} & \dots& a_{mn} \\
\end{bmatrix}
\tag {m x n阶矩阵}</script><h2 id="特殊矩阵"><a href="#特殊矩阵" class="headerlink" title="特殊矩阵"></a>特殊矩阵</h2><h3 id="行矩阵"><a href="#行矩阵" class="headerlink" title="行矩阵 :"></a>行矩阵 :</h3><script type="math/tex; mode=display">
A=\begin{bmatrix}
a_1&a_2&... & a_n\\
\end{bmatrix}</script><h3 id="列矩阵"><a href="#列矩阵" class="headerlink" title="列矩阵"></a>列矩阵</h3><script type="math/tex; mode=display">
A=\begin{bmatrix}
a_1 \\

a_2 \\
\vdots \\
a_n \\

\end{bmatrix}</script><h3 id="同形矩阵"><a href="#同形矩阵" class="headerlink" title="同形矩阵"></a>同形矩阵</h3><p>行数 列数相等</p>
<h3 id="零矩阵"><a href="#零矩阵" class="headerlink" title="零矩阵"></a>零矩阵</h3><h3 id="n阶方阵"><a href="#n阶方阵" class="headerlink" title="n阶方阵"></a>n阶方阵</h3><script type="math/tex; mode=display">
A=\begin{bmatrix}
a_{11}& a_{12} & \dots& a_{1n}\\
a_{21}& a_{22} & \dots& a_{2n}\\
\vdots& \vdots & \ddots&\vdots\\
a_{n1}& a_{m2} & \dots& a_{nn} \\
\end{bmatrix}
\tag {n阶矩阵}</script><h2 id="特殊矩阵-1"><a href="#特殊矩阵-1" class="headerlink" title="特殊矩阵"></a>特殊矩阵</h2><h3 id="上三角矩阵"><a href="#上三角矩阵" class="headerlink" title="上三角矩阵:"></a>上三角矩阵:</h3><script type="math/tex; mode=display">
A=\begin{bmatrix}
a_{11}& a_{12} & \dots& a_{1n}\\
0 & a_{22} & \dots& a_{2n}\\
\vdots& \vdots & \ddots&\vdots\\
0& 0 & \dots& a_{nn} \\
\end{bmatrix}
\tag {n阶矩阵}</script><h3 id="下三角矩阵"><a href="#下三角矩阵" class="headerlink" title="下三角矩阵:"></a>下三角矩阵:</h3><script type="math/tex; mode=display">
A=\begin{bmatrix}
a_{11}& 0 & \dots& 0\\
a_{21}& a_{22} & \dots& 0\\
\vdots& \vdots & \ddots&\vdots\\
a_{n1}& a_{n2} & \dots& a_{nn} \\
\end{bmatrix}
\tag {n阶矩阵}</script><h3 id="对角矩阵-A-n"><a href="#对角矩阵-A-n" class="headerlink" title="对角矩阵 $ A_n $"></a>对角矩阵 $ A_n $</h3><p>除对角线外全为0</p>
<script type="math/tex; mode=display">
A=\begin{bmatrix}
a_{11}& 0 & \dots& 0\\
0 & a_{22} & \dots& 0\\
\vdots& \vdots & \ddots&\vdots\\
0 & 0 & \dots& a_{nn} \\
\end{bmatrix}
\tag {n阶矩阵}</script><h3 id="单位矩阵-I-n"><a href="#单位矩阵-I-n" class="headerlink" title="单位矩阵 $ I_n $"></a>单位矩阵 $ I_n $</h3><p>对角线全为1的对角阵N阶对角方阵</p>
<script type="math/tex; mode=display">
I =\begin{bmatrix}
1 & 0 & \dots& 0\\
0 & 1 & \dots& 0\\
\vdots& \vdots & \ddots&\vdots\\
0& 0 & \dots& 1 \\
\end{bmatrix}
\tag {n阶方阵}</script><h2 id="增广炬阵"><a href="#增广炬阵" class="headerlink" title="增广炬阵"></a>增广炬阵</h2><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">线性方程组可以用增光矩阵表示</span><br></pre></td></tr></table></figure>
<p>系数矩阵$ A_n $, 常数项矩阵 b 并列组成的矩阵 C</p>
<p>C = [A, b]</p>
<h2 id="矩阵的初等行变换"><a href="#矩阵的初等行变换" class="headerlink" title="矩阵的初等行变换"></a>矩阵的初等行变换</h2><h2 id="方法"><a href="#方法" class="headerlink" title="方法"></a>方法</h2><ol>
<li>交换两行的位置</li>
<li>某行乘以常数K</li>
<li>把某一行的K倍加到另一行</li>
</ol>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">初等行变换是可逆的  称为 同解变换</span><br></pre></td></tr></table></figure>
<h2 id="两矩阵等价"><a href="#两矩阵等价" class="headerlink" title="两矩阵等价"></a>两矩阵等价</h2><p>A 初等行变换 变为B  : A B 等价</p>
<h1 id="利用MATLAB解方程组"><a href="#利用MATLAB解方程组" class="headerlink" title="利用MATLAB解方程组"></a>利用MATLAB解方程组</h1><h2 id="行最简型"><a href="#行最简型" class="headerlink" title="行最简型"></a>行最简型</h2><p>Reduced Row Echelon Form</p>
<p>把矩阵化为行最简形</p>
<p>行阶梯矩阵：逐行形成阶梯  </p>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">   <span class="number">9</span>    <span class="number">6</span>    <span class="number">2</span>   <span class="number">4</span></span><br><span class="line">   <span class="number">0</span>    <span class="number">3</span>    <span class="number">8</span>   <span class="number">7</span></span><br><span class="line">   <span class="number">0</span>    <span class="number">0</span>    <span class="number">5</span>   <span class="number">1</span></span><br><span class="line">   <span class="number">0</span>   	<span class="number">0</span>    <span class="number">0</span>   <span class="number">7</span></span><br><span class="line">   <span class="number">0</span>   	<span class="number">0</span>    <span class="number">0</span>   <span class="number">0</span></span><br></pre></td></tr></table></figure>
<p>行最简型<br>首个不为0的值为1</p>
<p>秩 4</p>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line"><span class="number">1</span>    <span class="number">6</span>    <span class="number">2</span>   <span class="number">4</span></span><br><span class="line"><span class="number">0</span>    <span class="number">1</span>    <span class="number">0</span>   <span class="number">7</span></span><br><span class="line"><span class="number">0</span>    <span class="number">0</span>    <span class="number">1</span>   <span class="number">7</span></span><br><span class="line"><span class="number">0</span>   	<span class="number">0</span>    <span class="number">0</span>   <span class="number">1</span></span><br><span class="line"><span class="number">0</span>   	<span class="number">0</span>    <span class="number">0</span>   <span class="number">0</span></span><br></pre></td></tr></table></figure>
<p>秩 3</p>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line"><span class="number">1</span>    <span class="number">6</span>    <span class="number">2</span>   <span class="number">4</span></span><br><span class="line"><span class="number">0</span>    <span class="number">1</span>    <span class="number">0</span>   <span class="number">7</span></span><br><span class="line"><span class="number">0</span>    <span class="number">0</span>    <span class="number">1</span>   <span class="number">7</span></span><br></pre></td></tr></table></figure>
<h2 id="rref-函数"><a href="#rref-函数" class="headerlink" title="rref 函数"></a>rref 函数</h2><ol>
<li>解线性方程</li>
<li>矩阵的秩  (几行非0)   有几个真正的约束条件</li>
<li>行最简形首元所在的列数</li>
</ol>
<p>行阶梯矩阵——&gt; 行最简形</p>
<p>例 1</p>
<script type="math/tex; mode=display">
A= \begin{bmatrix}
 2 & -2 & 2 & 6 \\
 2 & -1 & 2 & 4 \\
3 & -1 & 4 & 4 \\
1 & 1 & -1 & 3
\end{bmatrix}   
b = \begin{bmatrix}
-16 \\
-10 \\
-11 \\
-12
\end{bmatrix}</script><p>解：</p>
<p>ans = rref([A, b])</p>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; A=[<span class="number">2</span> <span class="number">-2</span> <span class="number">2</span> <span class="number">6</span>; <span class="number">2</span> <span class="number">-1</span> <span class="number">2</span> <span class="number">4</span>; <span class="number">3</span> <span class="number">-1</span> <span class="number">4</span> <span class="number">4</span>; <span class="number">1</span> <span class="number">1</span> <span class="number">-1</span> <span class="number">3</span>]</span><br><span class="line">A =</span><br><span class="line">     <span class="number">2</span>    <span class="number">-2</span>     <span class="number">2</span>     <span class="number">6</span></span><br><span class="line">     <span class="number">2</span>    <span class="number">-1</span>     <span class="number">2</span>     <span class="number">4</span></span><br><span class="line">     <span class="number">3</span>    <span class="number">-1</span>     <span class="number">4</span>     <span class="number">4</span></span><br><span class="line">     <span class="number">1</span>     <span class="number">1</span>    <span class="number">-1</span>     <span class="number">3</span></span><br><span class="line">&gt;&gt; b=[<span class="number">-16</span>; <span class="number">-10</span> ; <span class="number">-11</span>; <span class="number">-12</span>]</span><br><span class="line">b =</span><br><span class="line">   <span class="number">-16</span></span><br><span class="line">   <span class="number">-10</span></span><br><span class="line">   <span class="number">-11</span></span><br><span class="line">   <span class="number">-12</span></span><br><span class="line">&gt;&gt; rref([A,b])</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">     <span class="number">1</span>     <span class="number">0</span>     <span class="number">0</span>     <span class="number">0</span>    <span class="number">11</span></span><br><span class="line">     <span class="number">0</span>     <span class="number">1</span>     <span class="number">0</span>     <span class="number">0</span>    <span class="number">-8</span></span><br><span class="line">     <span class="number">0</span>     <span class="number">0</span>     <span class="number">1</span>     <span class="number">0</span>    <span class="number">-6</span></span><br><span class="line">     <span class="number">0</span>     <span class="number">0</span>     <span class="number">0</span>     <span class="number">1</span>    <span class="number">-7</span></span><br></pre></td></tr></table></figure>
<p>最后一列就是方程的解</p>
<p>例 2</p>
<p>判断解的性质和A的秩</p>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; A=[<span class="number">-2</span> <span class="number">-2</span>  <span class="number">2</span> <span class="number">2</span> <span class="number">-2</span>; <span class="number">1</span> <span class="number">-5</span> <span class="number">1</span> <span class="number">-3</span> <span class="number">-1</span>; <span class="number">-1</span> <span class="number">2</span> <span class="number">-5</span> <span class="number">6</span> <span class="number">5</span>; <span class="number">-1</span> <span class="number">2</span> <span class="number">1</span> <span class="number">0</span> <span class="number">-1</span>]</span><br><span class="line">A =</span><br><span class="line">    <span class="number">-2</span>    <span class="number">-2</span>     <span class="number">2</span>     <span class="number">2</span>    <span class="number">-2</span></span><br><span class="line">     <span class="number">1</span>    <span class="number">-5</span>     <span class="number">1</span>    <span class="number">-3</span>    <span class="number">-1</span></span><br><span class="line">    <span class="number">-1</span>     <span class="number">2</span>    <span class="number">-5</span>     <span class="number">6</span>     <span class="number">5</span></span><br><span class="line">    <span class="number">-1</span>     <span class="number">2</span>     <span class="number">1</span>     <span class="number">0</span>    <span class="number">-1</span></span><br><span class="line">&gt;&gt; b = [<span class="number">-2</span>; <span class="number">-1</span>; <span class="number">2</span> ; <span class="number">0</span>]</span><br><span class="line">b =</span><br><span class="line">    <span class="number">-2</span></span><br><span class="line">    <span class="number">-1</span></span><br><span class="line">     <span class="number">2</span></span><br><span class="line">     <span class="number">0</span></span><br><span class="line">&gt;&gt; rref([A,b])</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">    <span class="number">1.0000</span>         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>   <span class="number">-0.2222</span></span><br><span class="line">         <span class="number">0</span>    <span class="number">1.0000</span>         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>    <span class="number">0.2222</span></span><br><span class="line">         <span class="number">0</span>         <span class="number">0</span>    <span class="number">1.0000</span>         <span class="number">0</span>   <span class="number">-1.0000</span>   <span class="number">-0.6667</span></span><br><span class="line">         <span class="number">0</span>         <span class="number">0</span>         <span class="number">0</span>    <span class="number">1.0000</span>         <span class="number">0</span>   <span class="number">-0.3333</span></span><br></pre></td></tr></table></figure>
<p>故欠定 有无穷解  秩为4</p>
<p>ip = 1 2 3 4   不为0的首元</p>
<h1 id="应用实例"><a href="#应用实例" class="headerlink" title="应用实例"></a>应用实例</h1><h1 id=""><a href="#" class="headerlink" title="#"></a>#</h1><h2 id="平板材料温度问题"><a href="#平板材料温度问题" class="headerlink" title="平板材料温度问题"></a>平板材料温度问题</h2><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; A = [<span class="number">1</span> <span class="number">0</span> <span class="number">0</span> <span class="number">0</span>; <span class="number">1</span> <span class="number">1</span> <span class="number">1</span> <span class="number">1</span>; <span class="number">1</span> <span class="number">2</span> <span class="number">4</span> <span class="number">8</span>; <span class="number">1</span> <span class="number">3</span> <span class="number">9</span> <span class="number">27</span>]</span><br><span class="line">A =</span><br><span class="line">     <span class="number">1</span>     <span class="number">0</span>     <span class="number">0</span>     <span class="number">0</span></span><br><span class="line">     <span class="number">1</span>     <span class="number">1</span>     <span class="number">1</span>     <span class="number">1</span></span><br><span class="line">     <span class="number">1</span>     <span class="number">2</span>     <span class="number">4</span>     <span class="number">8</span></span><br><span class="line">     <span class="number">1</span>     <span class="number">3</span>     <span class="number">9</span>    <span class="number">27</span></span><br><span class="line">&gt;&gt; b = [<span class="number">3</span> ; <span class="number">0</span>; <span class="number">-1</span>; <span class="number">6</span>]</span><br><span class="line">b =</span><br><span class="line">     <span class="number">3</span></span><br><span class="line">     <span class="number">0</span></span><br><span class="line">    <span class="number">-1</span></span><br><span class="line">     <span class="number">6</span></span><br><span class="line">&gt;&gt; rref([A,b])</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">     <span class="number">1</span>     <span class="number">0</span>     <span class="number">0</span>     <span class="number">0</span>     <span class="number">3</span></span><br><span class="line">     <span class="number">0</span>     <span class="number">1</span>     <span class="number">0</span>     <span class="number">0</span>    <span class="number">-2</span></span><br><span class="line">     <span class="number">0</span>     <span class="number">0</span>     <span class="number">1</span>     <span class="number">0</span>    <span class="number">-2</span></span><br><span class="line">     <span class="number">0</span>     <span class="number">0</span>     <span class="number">0</span>     <span class="number">1</span>     <span class="number">1</span></span><br></pre></td></tr></table></figure>
<h2 id="交通节点车流量问题"><a href="#交通节点车流量问题" class="headerlink" title="交通节点车流量问题"></a>交通节点车流量问题</h2><h2 id="化学式配平问题"><a href="#化学式配平问题" class="headerlink" title="化学式配平问题"></a>化学式配平问题</h2><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; A = [3 0 -1 0; 8 0 0 -2; 0 2 -2 -1]</span><br><span class="line">A =</span><br><span class="line">       3              0             -1              0       </span><br><span class="line">       8              0              0             -2       </span><br><span class="line">       0              2             -2             -1       </span><br><span class="line">&gt;&gt; b = [0; 0; 0]</span><br><span class="line">b =</span><br><span class="line">       0       </span><br><span class="line">       0       </span><br><span class="line">       0       </span><br><span class="line">&gt;&gt; format rat, rref([A,b])</span><br><span class="line">ans =</span><br><span class="line">       1              0              0             -1/4            0       </span><br><span class="line">       0              1              0             -5/4            0       </span><br><span class="line">       0              0              1             -3/4            0</span><br></pre></td></tr></table></figure>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">1. 了解二，三阶线性方程组的图形和几何意义</span><br><span class="line">2. 利用初等变换把 增广矩阵 转化为 行最简形</span><br><span class="line">3. 秩表明独立方程的个数</span><br><span class="line">4. 系数矩阵和增广矩阵的秩相等是线性方程组有解的充分必要条件</span><br></pre></td></tr></table></figure>

      
    </div>
    <footer class="article-footer">
      <a data-url="http://yoursite.com/2019/01/14/线性代数/实用大众线性代数/第1周线性方程组与矩阵/" data-id="cjz7utmp00021xhpeg0wao1wp" class="article-share-link">分享</a>
      
      
      
  <ul class="article-tag-list"><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/rref/">rref</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/几何意义/">几何意义</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/初等行变换/">初等行变换</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/增广矩阵/">增广矩阵</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/矩阵的秩/">矩阵的秩</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/线性方程组/">线性方程组</a></li></ul>

    </footer>
  </div>
  
</article>
 


  


  <nav id="page-nav">
    <a class="extend prev" rel="prev" href="/page/4/">&laquo; 上一页</a><a class="page-number" href="/">1</a><span class="space">&hellip;</span><a class="page-number" href="/page/3/">3</a><a class="page-number" href="/page/4/">4</a><span class="page-number current">5</span>
  </nav>
</section>
           
    <aside id="sidebar">
  
    

  
    
  
    

  
    
  
    
  <div class="widget-wrap">
    <h3 class="widget-title recent-posts">最新文章</h3>
    <div class="widget">
      <ul>
        
          <li>
            <a href="/2019/08/12/数据结构与算法设计/小甲鱼/查找/">查找</a>
          </li>
        
          <li>
            <a href="/2019/08/05/数据结构与算法设计/小甲鱼/二叉树/">二叉树</a>
          </li>
        
          <li>
            <a href="/2019/08/05/数据结构与算法设计/小甲鱼/树/">树</a>
          </li>
        
          <li>
            <a href="/2019/07/29/English/Writer/money/">money</a>
          </li>
        
          <li>
            <a href="/2019/07/29/English/Words/theme-words/">theme_words</a>
          </li>
        
      </ul>
    </div>
  </div>

  
    

  
    
  
    <!--微信公众号二维码-->


  
</aside>

      </div>
      <footer id="footer">
  
  <div class="outer">
    <div id="footer-left">
      &copy; 2014 - 2019 tiger&nbsp;|&nbsp;
      主题 <a href="https://github.com/giscafer/hexo-theme-cafe/" target="_blank">Cafe</a>
    </div>
     <div id="footer-right">
      联系方式&nbsp;|&nbsp;375478250@qq.com
    </div>
  </div>
</footer>
 <script src="/jquery/jquery.min.js"></script>

 <script src='https://unpkg.com/mermaid@8.1.0/dist/mermaid.min.js'></script>
 <script>
 if (window.mermaid) {
	 mermaid.initialize({startOnLoad:true});
 }
</script>

    </div>
    <nav id="mobile-nav">
  
    <a href="/" class="mobile-nav-link">首页</a>
  
    <a href="/archives" class="mobile-nav-link">归档</a>
  
    <a href="/categories/高等数学" class="mobile-nav-link">高等数学</a>
  
    <a href="/categories/线性代数" class="mobile-nav-link">线性代数</a>
  
    <a href="/categories/数据结构与算法" class="mobile-nav-link">数据结构与算法</a>
  
    <a href="/categories/English" class="mobile-nav-link">英语</a>
  
    <a href="/about" class="mobile-nav-link">关于</a>
  
</nav>
    <img class="back-to-top-btn" src="/images/fly-to-top.png"/>
<script>
// Elevator script included on the page, already.
window.onload = function() {
  var elevator = new Elevator({
    selector:'.back-to-top-btn',
    element: document.querySelector('.back-to-top-btn'),
    duration: 1000 // milliseconds
  });
}
</script>
    <!-- author:forvoid begin -->
<!-- author:forvoid end -->


  
    <script type="text/x-mathjax-config">
      MathJax.Hub.Config({
        tex2jax: {
          inlineMath: [ ['$','$'], ["\\(","\\)"]  ],
          processEscapes: true,
          skipTags: ['script', 'noscript', 'style', 'textarea', 'pre', 'code']
        }
      })
    </script>

    <script type="text/x-mathjax-config">
      MathJax.Hub.Queue(function() {
        var all = MathJax.Hub.getAllJax(), i;
        for (i=0; i < all.length; i += 1) {
          all[i].SourceElement().parentNode.className += ' has-jax';
        }
      })
    </script>
    <script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML"></script>
  


 <script src="/js/is.js"></script>


  <link rel="stylesheet" href="/fancybox/jquery.fancybox.css">
  <script src="/fancybox/jquery.fancybox.pack.js"></script>


<script src="/js/script.js"></script>
<script src="/js/elevator.js"></script>
  </div>
</body>
</html>
